Old and new results on packing arborescences Zolt an Szigeti - PowerPoint PPT Presentation

Old and new results on packing arborescences Zolt´ an Szigeti ´ Equipe Optimisation Combinatoire Laboratoire G-SCOP INP Grenoble, France 11 juin 2015 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 1 / 26

Outline Old results Digraphs Packing spanning arborescences Packing maximal arborescences Dypergraphs Packing spanning hyper-arborescences Packing maximal hyper-arborescences Matroid-based rooted-digraphs Matroid-based packing of rooted-arborescences Maximal-rank packing of rooted-arborescences New results Matroid-based rooted-dypergraph Matroid-based packing of rooted-hyper-arborescences Maximal-rank packing of rooted-hyper-arborescences Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 2 / 26

Reachability in digraph Definition Let � G = ( V , A ) be a digraph and X ⊆ V . 1 ρ A ( X ) is the number of arcs entering X , 2 P A ( X ) is the set of vertices from which X can be reached in � G , 3 Q A ( X ) is the set of vertices that can be reached from X in � G . ρ A ( X ) = 2 u X x ′ v x P A ( X ) Q A ( X ) V \ X X � G Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 3 / 26

Arborescences Definition Let � G = ( V , A ) be a digraph and r ∈ V . 1 A subgraph � T = ( U , B ) of � G is an r -arborescence if r 1 r ∈ U with ρ B ( r ) = 0 , 1 ρ B ( u ) = 1 for all u ∈ U \ r and � 2 T 1 ρ B ( X ) ≥ 1 for all X ⊆ V \ r , 3 � X ∩ U � = ∅ . T 2 2 An r -arborescence � T is r 2 spanning if U = V , 1 maximal if U = Q A ( r ). 2 3 Packing of arborescences is a set of pairwise arc-disjoint arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 4 / 26

Packing spanning arborescences Theorem (Edmonds 1973) Let � G = ( V , A ) be a digraph, r ∈ V and k a positive integer. 1 There exists a packing of k spanning r-arborescences ⇐ ⇒ 2 ρ A ( X ) ≥ k for all ∅ � = X ⊆ V \ r . � � T 1 T 2 r Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 5 / 26

Packing maximal arborescences Definition Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 A packing of maximal arborescences is a set { � T 1 , . . . , � T t } of pairwise arc-disjoint maximal r i -arborescences � T i in � G ; that is for every v ∈ V , { r i : v ∈ V ( � T i ) } = { r i ∈ P A ( v ) } . 2 For X ⊆ V , p A ( X ) = |{ r i ∈ P A ( X ) \ X }| . r 1 r 1 � T 1 p A ( X ) = 2 � T 2 X r 2 r 2 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 6 / 26

Packing maximal arborescences Theorem (Kamiyama, Katoh, Takizawa 2009) Let � G = ( V , A ) be a digraph and ( r 1 , . . . , r t ) ∈ V t . 1 There exists a packing of maximal arborescences ⇐ ⇒ 2 ρ A ( X ) ≥ p A ( X ) for all X ⊆ V . Remark It implies Edmonds’ theorem. 1 Let r 1 = · · · = r k = r . 2 ρ A ( X ) ≥ k for all ∅ � = X ⊆ V \ r implies the above condition and that each vertex is reachable from r . 3 Hence there exists a packing of maximal r -arborescences that is a packing of spanning r -arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 7 / 26

Dypergraphs Definition 1 Directed hypergraph (shortly dypergraph) is � G = ( V , A ), where V denotes the set of vertices and A denotes the set of hyperarcs of � G . 2 Hyperarc is a pair ( Z , z ) such that z ∈ Z ⊆ V , where z is the head of the hyperarc ( Z , z ) and the elements of Z \ z � = ∅ are the tails of the hyperarc ( Z , z ) . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 8 / 26

Reachability in dypergraph Definition Let � G = ( V , A ) be a dypergraph and X ⊆ V . 1 Hyperarc ( Z , z ) enters X if z ∈ X and ( Z \ z ) ∩ ( V \ X ) � = ∅ , 2 ρ A ( X ) is the number of hyperarcs entering X , 3 Path from u to x in � G is v 1 (= u ) , ( Z 1 , v 2 ) , v 2 , . . . , v i , ( Z i , v i +1 ) , v i +1 , . . . , v j (= x ) such that v i is a tail of ( Z i , v i +1 ) . 4 P A ( X ) is the set of vertices from which X can be reached in � G , 5 Q A ( X ) is the set of vertices that can be reached from X in � G . ρ A ( X ) = 2 u x v X x ′ X V \ X P A ( X ) Q A ( X ) � G Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 9 / 26

Trimming Definition Trimming the dypergraph � G means trimming replacing each hyperarc ( K , v ) of � G by an arc uv where u is one of the tails of the hyperarc ( K , v ). Definition h is supermodular : h ( X ) + h ( Y ) ≤ h ( X ∩ Y ) + h ( X ∪ Y ) ∀ X , Y ⊆ V . Theorem (Frank 2011) Let � G = ( V , A ) be a dypergraph and h an integer-valued, intersecting supermodular function on V such that h ( ∅ ) = 0 = h ( V ) . G can be trimmed to a digraph � If ρ A ( X ) ≥ h ( X ) for all X ⊆ V , then � G such that ρ A ( X ) ≥ h ( X ) for all X ⊆ V . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 10 / 26

Hyper-arborescences Definition Let � G = ( V , A ) be a dypergraph and r ∈ V . � T 1 r 1 1 A subgraph � T = ( U , B ) of � G is an U ∗ 2 r -hyper-arborescence if it can be trimmed to an r -arborescence on U ∗ ∪ r , where r 2 � U ∗ = { u : ρ B ( u ) � = 0 } ; that is T 2 r ∈ U \ U ∗ , 1 ρ B ( u ) = 1 for all u ∈ U ∗ and 2 r 1 ρ B ( X ) ≥ 1 for all X ⊆ V \ r , 3 X ∩ U ∗ � = ∅ . � T 1 2 The r -hyper-arborescence � T is � T 2 spanning if U ∗ = V \ r , 1 r 2 maximal if U ∗ = Q A ( r ) \ r . 2 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 11 / 26

Packing spanning hyper-arborescences Theorem (Frank, T. Kir´ aly, Kriesell 2003) Let � G = ( V , A ) be a dypergraph, r ∈ V and k a positive integer. 1 There exists a packing of k spanning r-hyper-arborescences ⇐ ⇒ 2 ρ A ( X ) ≥ k for all ∅ � = X ⊆ V \ r . Remark 1 It is proved easily by trimming and Edmonds’ theorem. 2 It implies Edmonds’ theorem if � G is a digraph. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 12 / 26

Packing maximal hyper-arborescences Theorem (B´ erczi, Frank 2008) Let � G = ( V , A ) be a dypergraph and ( r 1 , . . . , r t ) ∈ V t . 1 There exists a packing of maximal hyper-arborescences ⇐ ⇒ 2 ρ A ( X ) ≥ p A ( X ) for all X ⊆ V . Remark 1 It is proved not easily by trimming and Kamiyama, Katoh, Takizawa’s theorem since p A ( X ) is not intersecting supermodular. 2 It implies Frank, T. Kir´ aly, Kriesell’s theorem if r 1 = · · · = r k = r and ρ A ( X ) ≥ k 1 for all ∅ � = X ⊆ V \ r , Kamiyama, Katoh, Takizawa’s theorem if � G is a digraph. 2 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 13 / 26

Matroids Definition For I ⊆ 2 S , M = (S , I ) is a matroid if 1 I � = ∅ , 2 If X ⊆ Y ∈ I then X ∈ I , 3 If X , Y ∈ I with | X | < | Y | then ∃ y ∈ Y \ X such that X ∪ y ∈ I . Examples 1 Sets of linearly independent vectors in a vector space, 2 Edge-sets of forests of a graph, 3 U n , k = { X ⊆ S : | X | ≤ k } where | S | = n , free matroid = U n , n . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 14 / 26

Matroids Notion 1 independent sets = I , any subset of an independent set is independent, 1 2 base = maximal independent set, all basis are of the same size, 1 3 rank function : r ( X ) = max {| Y | : Y ∈ I , Y ⊆ X } . non-decreasing, 1 submodular (that is − r is supermodular), 2 X ∈ I if and only if r ( X ) = | X | . 3 Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 15 / 26

Matroid-based rooted-digraphs Definition A matroid-based rooted-digraph is a quadruple ( � G , M , S , π ) : � G = ( V , A ) is a digraph, 1 2 M is a matroid on a set S = { s 1 , . . . , s t } . 3 π is a placement of the elements of S at vertices of V such that S v ∈ I for every v ∈ V , where S X = π − 1 ( X ) , the elements of S placed at X . π ( s 1 ) π ( s 1 ) X π ( s 2 ) π ( s 2 ) � S X = { s 1 , s 2 } G S = { s 1 , s 2 , s 3 } M = U 3 , 2 π ( s 3 ) π ( s 3 ) Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 16 / 26

Matroid-based packing of rooted-arborescences Definition π ( s 1 ) π ( s 2 ) A rooted-arborescence is a pair ( � T , s) where T 2 � T is an r -arborescence for some vertex r , 1 T 1 2 s ∈ S, placed at r . T 3 π ( s 3 ) Definition A packing { ( � T 1 , s 1 ) , . . . , ( � T | S | , s | S | ) } of rooted-arborescences is matroid-based if { s i ∈ S : v ∈ V ( � T i ) } forms a base of S for every v ∈ V . Remark For the free matroid M with all k roots at a vertex r , 1 matroid-based packing of rooted-arborescences ⇐ ⇒ 2 packing of k spanning r -arborescences. Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 17 / 26

Matroid-based packing of rooted-arborescences Theorem (Durand de Gevigney, Nguyen, Szigeti 2013) Let ( � G , M , S , π ) be a matroid-based rooted-digraph. 1 There is a matroid-based packing of rooted-arborescences ⇐ ⇒ 2 ρ A ( X ) ≥ r M (S) − r M (S X ) for all ∅ � = X ⊆ V . Remark It implies Edmonds’ theorem if M is the free matroid with all k roots at the vertex r . Z. Szigeti (G-SCOP, Grenoble) On packing of arborescences 11 juin 2015 18 / 26